We aim to calculate the first and second Stiefel Whitney classes of orthogonally irreducible representations of \(G\) .
The paper [1]} gives criteria to determine spinorial representations of \(G\) . We know that an orthogonal representation \(\pi \)  of \(G\)  is spinorial iff \(w_2(\pi )=w_1(\pi )\cup w_1(\pi )\)  (see [2]}). Moreover \(H^2(GL_2(\mathbb {F}_q), \mathbb {Z}/2\mathbb {Z}) = \mathbb {Z}/2\mathbb {Z}\) . Hence if we know \(w_1\)  we also can derive \(w_2\) .
